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DOI: 10.4230/LIPIcs.SoCG.2025.14

Extremal Betti Numbers and Persistence in Flag Complexes

Authors: Lies Beers and Magnus Bakke Botnan

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract

We investigate several problems concerning extremal Betti numbers and persistence in filtrations of flag complexes. For graphs on n vertices, we show that β_k(X(G)) is maximal when G = 𝒯_{n,k+1}, the Turán graph on k+1 partition classes, where X(G) denotes the flag complex of G. Building on this, we construct an edgewise (one edge at a time) filtration 𝒢 = G₁ ⊆ ⋯ ⊆ 𝒯_{n,k+1} for which β_k(X(G_i)) is maximal for all graphs on n vertices and i edges. Moreover, the persistence barcode ℬ_k(X(G)) achieves a maximal number of intervals, and total persistence, among all edgewise filtrations with |E(𝒯_{n,k+1})| edges. For k = 1, we consider edgewise filtrations of the complete graph K_n. We show that the maximal number of intervals in the persistence barcode is obtained precisely when G_{⌈n/2⌉ ⋅ ⌊n/2⌋} = 𝒯_{n,2}. Among such filtrations, we characterize those achieving maximal total persistence. We further show that no filtration can optimize β₁(X(G_i)) for all i, and conjecture that our filtrations maximize the total persistence over all edgewise filtrations of K_n.

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Lies Beers and Magnus Bakke Botnan. Extremal Betti Numbers and Persistence in Flag Complexes. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 14:1-14:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{beers_et_al:LIPIcs.SoCG.2025.14,
  author =	{Beers, Lies and Bakke Botnan, Magnus},
  title =	{{Extremal Betti Numbers and Persistence in Flag Complexes}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{14:1--14:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.14},
  URN =		{urn:nbn:de:0030-drops-231668},
  doi =		{10.4230/LIPIcs.SoCG.2025.14},
  annote =	{Keywords: Topological data analysis, Extremal graph theory}
}

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Extremal Betti Numbers and Persistence in Flag Complexes

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